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Theorem ltrncnvatb 30253
Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)
Hypotheses
Ref Expression
ltrnatb.b  |-  B  =  ( Base `  K
)
ltrnatb.a  |-  A  =  ( Atoms `  K )
ltrnatb.h  |-  H  =  ( LHyp `  K
)
ltrnatb.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrncnvatb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A
) )

Proof of Theorem ltrncnvatb
StepHypRef Expression
1 ltrnatb.b . . . . . 6  |-  B  =  ( Base `  K
)
2 ltrnatb.h . . . . . 6  |-  H  =  ( LHyp `  K
)
3 ltrnatb.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3ltrn1o 30239 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
543adant3 977 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  F : B
-1-1-onto-> B )
6 simp3 959 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  P  e.  B )
7 f1ocnvdm 5958 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  P  e.  B )  ->  ( `' F `  P )  e.  B
)
85, 6, 7syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( `' F `  P )  e.  B )
9 ltrnatb.a . . . 4  |-  A  =  ( Atoms `  K )
101, 9, 2, 3ltrnatb 30252 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( `' F `  P )  e.  B
)  ->  ( ( `' F `  P )  e.  A  <->  ( F `  ( `' F `  P ) )  e.  A ) )
118, 10syld3an3 1229 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( `' F `  P )  e.  A  <->  ( F `  ( `' F `  P ) )  e.  A ) )
12 f1ocnvfv2 5955 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  P  e.  B )  ->  ( F `  ( `' F `  P ) )  =  P )
135, 6, 12syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( F `  ( `' F `  P ) )  =  P )
1413eleq1d 2454 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( F `  ( `' F `  P )
)  e.  A  <->  P  e.  A ) )
1511, 14bitr2d 246 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   `'ccnv 4818   -1-1-onto->wf1o 5394   ` cfv 5395   Basecbs 13397   Atomscatm 29379   HLchlt 29466   LHypclh 30099   LTrncltrn 30216
This theorem is referenced by:  ltrncnvat  30256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-undef 6480  df-riota 6486  df-map 6957  df-plt 14343  df-glb 14360  df-p0 14396  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-hlat 29467  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220
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